In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations

Unlike the conjugate gradient method, this algorithm does not require the matrix to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose .

The Algorithm

1. and a preconditioner

do

satisfy

and thus are the respective residuals, as approximate solutions to the systems

is the adjoint is the complex conjugate.

Unpreconditioned version of the algorithm

1. ,

do

Discussion

The biconjugate gradient method is numerically unstable (compare to the biconjugate gradient stabilized method), but very important from a theoretical point of view. Define the iteration steps by

using the related projection

with

These related projections may be iterated themselves as

A relation to Quasi-Newton methods, where

The new directions

are then orthogonal to the residuals:

which themselves satisfy

.

The biconjugate gradient method now makes a special choice and uses the setting

and are avoided, and the algorithm takes the form stated above.

Properties

• is self-adjoint, and the conjugate gradient method at half the computational cost.

• The sequences produced by the algorithm are biorthogonal.

• . The algorithm thus produces projections onto the Krylov subspace.

• .

See also

• Biconjugate gradient stabilized method (BiCG-Stab)

• Conjugate gradient method (CG)

• Conjugate gradient squared method (CGS)

References